QCD sum rule analysis of Heavy Quarkonium states in magnetized matter -- effects of (inverse) magnetic catalysis

The masses of the $1S$ and $1P$ states of heavy quarkonia are investigated in the magnetized, asymmetric nuclear medium, accounting for the Dirac sea effects, using a combined approach of chiral effective model and QCD sum rule method. These are calculated from the in-medium scalar and twist-2 gluon condensates, calculated within the chiral model. The gluon condensate is simulated through the scalar dilaton field, $\chi$ introduced in the model through a scale-invariance breaking logarithmic potential. Considering the scalar fields to be classical, the dilaton field, $\chi$, the non-strange isoscalar, $\sigma (\sim (\langle \bar u u\rangle +\langle \bar d d\rangle ))$, strange isoscalar, $\zeta (\sim \langle \bar s s\rangle)$ and non-strange isovector, $\delta (\sim (\langle\bar u u\rangle-\langle\bar d d\rangle)$) fields, are obtained by solving their coupled equations of motion, as derived from the chiral model Lagrangian. The effects of magnetic field due to the Dirac sea as well as the Landau energy levels of protons, and the non-zero anomalous magnetic moments of the nucleons are considered in the present study. In presence of an external magnetic field, there is also mixing between the longitudinal component of the vector meson and pseudoscalar meson (PV mixing) in both quarkonia sectors, leading to a rise (drop) of the masses of $J/\psi^{||}\ (\eta_c$) and $\Upsilon^{||}(1S)\ (\eta_b$) states. These might show in the experimental observables, e.g., the dilepton spectra in the non-central, ultra-relativistic heavy ion collision experiments at RHIC and LHC, where the produced magnetic field is huge.